<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution</dfn> Seek a solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y_2=v(x) y_1=v(x) \sin x^2.
\end{equation*}
</div>
<p class="continuation">Substituting it into the ODE, one has</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
&amp; x [v^{\prime \prime} y_1+2 v^{\prime} y_1^{\prime}+v y_1^{\prime \prime}]-(v^{\prime} y_1+v y_1^{\prime}) +4 x^3 v y_1=0\\
&amp;\to~ v [ x y_1^{\prime \prime} -y_1^{\prime}+4 x^3 y_1]+x y_1 v^{\prime \prime}+v^{\prime} [2 x y_1^{\prime}-y_1]=0\\
&amp;\to~\qquad \qquad \qquad \quad 0+x y_1 v^{\prime \prime}+v^{\prime} [2 x y_1^{\prime}-y_1]=0.
\end{aligned}
\end{equation*}
</div>
<p class="continuation">Let <span class="process-math">\(u=\frac{\textrm{d} v}{\textrm{d} x}\text{,}\)</span> we have</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
&amp;x \sin x^2 \frac{\textrm{d} u}{\textrm{d} x}+[4 x^2 \cos x^2-\sin x^2 ] u=0\\
&amp;\to~\frac{\textrm{d} u}{u}=-\frac{4 x^2 \cos x^2-\sin x^2}{x \sin x^2}  \textrm{d} x=-\frac{4 x \cos x^2}{\sin x^2} \textrm{d} x+\frac{1}{x} \textrm{d} x\\
&amp;\ln |u|=-\int \frac{2 \textrm{d} \sin x^2}{\sin x^2}+\ln |x|+C_1=-2 \ln |\sin x^2|+\ln |x|+C_1\\
&amp;\to~u=C_2 \frac{x}{(\sin x^2)^2}.
\end{aligned}
\end{equation*}
</div>
<span class="incontext"><a href="sec3_5.html#p-111" class="internal">in-context</a></span>
</body>
</html>
